Set \(t:=\operatorname{depth}R-\operatorname{depth}_R(M)\) . As \(M\)  has a bounded Betti numbers, by [1]}, the minimal resolution of
\(\Omega ^{t}M\)  is periodic of period at most two, hence so is the minimal complete resolution of \(M\) . Therefore, we get the following isomorphisms:
\(\widehat{\operatorname{Tor}}_i^R(M,N)\cong \widehat{\operatorname{Tor}}_{i+2}^R(M,N) \text{ for all } i\in \mathbb {Z}.\qquad \mathrm {(\ref {cx1}.1)}\) 
Note that by Theorem REF  and assumption (ii), \(\operatorname{Supp}_R(\widehat{\operatorname{Tor}}_i^R(M,N))\subseteq \operatorname{\mathcal {Z}}\)  for all \(i\in \mathbb {Z}\) . Now the first assertion follows from Theorem REF (i) and (REF .1). To prove the second assertion, suppose that \(\operatorname{H}^n_{\operatorname{\mathcal {Z}}}(M\otimes _RN)=0\) . By Theorem REF (ii), \(\widehat{\operatorname{Tor}}_{-n}^R(M,N)=0\)  and so by (REF .1), \(\widehat{\operatorname{Tor}}_{-n+2j}^R(M,N)=0\)  for all \(j\in \mathbb {Z}\) . Therefore, by using Theorem REF (i) we see that \(\operatorname{H}^{n-2i}_{\operatorname{\mathcal {Z}}}(M\otimes _RN)=0\)  for all integer \(i\ge 0\) .
